To determine the domain of the function [tex]\( f(x) = 3^{x-2} \)[/tex], we need to analyze for which values of [tex]\( x \)[/tex] the function is defined.
1. Understanding the function:
The given function is [tex]\( f(x) = 3^{x-2} \)[/tex]. This is an exponential function where the base is 3, and the exponent is [tex]\( x-2 \)[/tex].
2. Characteristics of exponential functions:
Exponential functions of the form [tex]\( a^{x} \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on [tex]\( x \)[/tex] because you can raise a positive real number to any real power.
3. Formulating the domain:
Since 3 is a positive real number and there are no restrictions on the exponent [tex]\( x-2 \)[/tex], [tex]\( 3^{x-2} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
4. Conclusion:
The domain of the function [tex]\( f(x) = 3^{x-2} \)[/tex] consists of all real numbers. Therefore, the correct description of the domain is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
Thus, the correct answer is:
[tex]\[
\{ x \mid x \text{ is a real number} \}
\][/tex]
